easy derivative but it took me 32 minutes

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  • เผยแพร่เมื่อ 22 พ.ย. 2024

ความคิดเห็น • 270

  • @blackpenredpen
    @blackpenredpen  ปีที่แล้ว +111

    If this one limit isn't crazy enough, then try 100 limits: th-cam.com/video/TglD4Y6lmQk/w-d-xo.htmlsi=lR_jfa-gI8FVO7VS

    • @muddle.
      @muddle. ปีที่แล้ว +5

      already done sir.

    • @Mandq.
      @Mandq. ปีที่แล้ว +2

      easy 🙅🏼‍♀

    • @francorota8638
      @francorota8638 11 หลายเดือนก่อน

      This is the first time I was introduced to symmetric derivatives, as my college professors never taught me about it.

  • @mathmachine4266
    @mathmachine4266 ปีที่แล้ว +394

    Unfortunately, this problem is unsolvable. Not because there is no solution, but because my board isn't big enough.

    • @TomFarrell-p9z
      @TomFarrell-p9z ปีที่แล้ว +27

      Why, it's a one line derivation! If you start sufficiently far to the left. 🙂

    • @unanimatereactor5014
      @unanimatereactor5014 ปีที่แล้ว

      ​@@TomFarrell-p9zas long as the size of your letter approaches 0

    • @senseof_outrage9390
      @senseof_outrage9390 11 หลายเดือนก่อน +22

      I have discovered a truly marvelous demonstration of this proposition however this whiteboard is too small to contain.

    • @thexavier666
      @thexavier666 10 หลายเดือนก่อน +3

      Assume you have an infinite whiteboard

    • @unanimatereactor5014
      @unanimatereactor5014 10 หลายเดือนก่อน +3

      @@thexavier666 no it tends to infinity

  • @YoungPhysicistsClub1729
    @YoungPhysicistsClub1729 ปีที่แล้ว +884

    Bro really used the limit definition, legend status

    • @blackpenredpen
      @blackpenredpen  ปีที่แล้ว +106

      😆

    • @thomasblackwell9507
      @thomasblackwell9507 ปีที่แล้ว +21

      That is a fact!

    • @bryantwiltrout5492
      @bryantwiltrout5492 ปีที่แล้ว +30

      Pure madness too 😂😂😂 using the Limit Definition to do derivatives of Trig functions is crazy 😂😂

    • @andreasxfjd4141
      @andreasxfjd4141 11 หลายเดือนก่อน +2

      Even worse to integrate with series

    • @lukeparsons4965
      @lukeparsons4965 10 หลายเดือนก่อน +1

      I’m just gonna do a reihman sum with infinite rectangles

  • @mcalkis5771
    @mcalkis5771 ปีที่แล้ว +462

    Now you have to do an epsilon delta proof of the limit for the ultimate presentation of mathematical rigor.

    • @blackpenredpen
      @blackpenredpen  ปีที่แล้ว +95

      😂

    • @yttyw8531
      @yttyw8531 ปีที่แล้ว +50

      but before that you need to proof that 1+1=2

    • @MrFeast-l1d
      @MrFeast-l1d ปีที่แล้ว +20

      This would take 12 hours but would be a good video

    • @Kanin105
      @Kanin105 11 หลายเดือนก่อน +1

      Xddddddddddd

    • @adb012
      @adb012 11 หลายเดือนก่อน +4

      @@blackpenredpen ... Why do you laugh? It was not a joke.

  • @apolloo9068
    @apolloo9068 ปีที่แล้ว +83

    It's reassuring that I'm not the only one prone to making calculation errors. Great video!

  • @Romeo-qk8tk
    @Romeo-qk8tk ปีที่แล้ว +214

    As an aspiring student in AP Calculus, this video was incredible to see! Awesome content! ❤

    • @blackpenredpen
      @blackpenredpen  ปีที่แล้ว +25

      Thank you!

    • @Sir_Isaac_Newton_
      @Sir_Isaac_Newton_ ปีที่แล้ว +41

      You're gonna fail AP Calculus blud

    • @harrymetu2746
      @harrymetu2746 ปีที่แล้ว +2

      ​@@Sir_Isaac_Newton_😂😂😂😂

    • @harrymetu2746
      @harrymetu2746 ปีที่แล้ว +6

      ​@@Sir_Isaac_Newton_when Newton says it 💀

    • @obi-wankenobigoat
      @obi-wankenobigoat 11 หลายเดือนก่อน +7

      @@Sir_Isaac_Newton_I just got an A on my semester final it’s really not that hard if you pay attention (assuming good teacher)

  • @dinohuntr851
    @dinohuntr851 8 หลายเดือนก่อน +4

    You have become my favorite TH-camr. Your teaching style is fun, you aren't afraid to show us your mistakes, and you are just enjoyable to watch. I can tell you genuinely want to teach, not just show off your skills. Keep up the good work!!

  • @laurensdehaan2202
    @laurensdehaan2202 11 หลายเดือนก่อน +15

    Man, your enthusiasm is SO contagious! I stood up here in front of my computer and watched the whole thing straight through, with a couple of pauses to reassure myself why some things worked out the way they did! Thank you!

    • @EmpyreanLightASMR
      @EmpyreanLightASMR 11 หลายเดือนก่อน +1

      Same! My attention span is such that I can watch a full movie over the course of several days. I saw this video and thought I'd kill a few minutes before going to make dinner and ended up watching the whole thing. I was rapt!

  • @paytonholmes6019
    @paytonholmes6019 ปีที่แล้ว +113

    I don’t think I was ever taught what a symmetrical derivative is in my calculus classes. Thank you.

    • @fabianwho9797
      @fabianwho9797 11 หลายเดือนก่อน

      Im no expert, but in my judgement it is rarely useful for anything, so most people never hear of it

    • @syrup767
      @syrup767 11 หลายเดือนก่อน

      @@fabianwho9797 you can use it in numerical applications as a second order approximation of the derivative: [f(x+h)-f(x-h)]/(2h) = f'(x) + O(h²) in contrast to [f(x+h)-f(x)]/h = f'(h) + O(h) (for h→0, assuming f∈C²)

    • @EmpyreanLightASMR
      @EmpyreanLightASMR 11 หลายเดือนก่อน +2

      I'm still a bit lost as to why he had to go that route. Maybe he said why and I missed it. The symmetrical derivative can be used for a symmetrical function about a non-differentiable point, I guess. But why does he use it for sin(x^2)?

    • @bred223
      @bred223 9 หลายเดือนก่อน +3

      @@EmpyreanLightASMRusing the normal def of deriv i got it in about 7 minutes so i assume it was just for funsies

    • @maxmustermann3938
      @maxmustermann3938 8 หลายเดือนก่อน +5

      ​​@@fabianwho9797 it is very commonly used for numeric derivatives (Central differencing), especially on i.e. images when computing gradients or laplacians or calculating the slope or the normal of a heightmap, also heavily used to solve grid-based fluid simulations

  • @jimschneider799
    @jimschneider799 8 หลายเดือนก่อน +4

    I'm neither a student nor a teacher. I'm just an old fart engineer who loves math enough to realize I really let myself get rusty on the basics. So, although most of the math I do at work is related to number theory, I do appreciate the refresher, particularly since you tend to tackle problems in ways different than what I would use.

  • @happyhippo4664
    @happyhippo4664 ปีที่แล้ว +175

    I am a 64 year old chemical engineer, still working. Math has always been my strongest subject. I enjoy these videos very much. I feel if you do not understand math that well, you will have lot more difficulty in engineering.

    • @blackpenredpen
      @blackpenredpen  ปีที่แล้ว +34

      Thank you!!

    • @mesindetrabajinv666
      @mesindetrabajinv666 ปีที่แล้ว +1

      Can you give me some advice for college and engineering? I’m planning to study Aerospace engineering

    • @Scorik375
      @Scorik375 ปีที่แล้ว +1

      and even in many aspects of life

    • @happyhippo4664
      @happyhippo4664 ปีที่แล้ว

      @@mesindetrabajinv666 Look at the occupational handbook for job outlook. Chem Engineering was hard but I've heard aerospace is even harder. My concern is, unless you are exceptional, it may be harder t find a job in that field. I started in Chemistry but switched to Chemical Engineering when I found out that BS ChEs were getting almost same pay as PhD Chemists. Probably more important, do what you enjoy.

    • @exodiara6392
      @exodiara6392 ปีที่แล้ว

      Im not so sure. Im not applying so much advanced mathematics again in carrier. Even friends that were more advanced had forgotten linear algebra.

  • @slytherinbrian
    @slytherinbrian ปีที่แล้ว +9

    This is better than anything on netflix!

  • @JoaoVictorCavalcanteMiranda
    @JoaoVictorCavalcanteMiranda 8 หลายเดือนก่อน +3

    Thanks for the great video!
    I didn't know about symmetrical derivatives until now!
    Your excitement is worth of the challenge!

  • @jaysonbunnell8097
    @jaysonbunnell8097 ปีที่แล้ว +15

    This was super awesome! I took calc 1 in highschool, and I've taken calc 2, Differential Equations, and Linear Algebra in college. I don't have many math credits left to take, so I find these videos awesome for keeping me on my math skills. Thank you!!

  • @바다노을-o3r
    @바다노을-o3r 11 หลายเดือนก่อน +3

    Sinx^2을 두번 미분~
    첫번째미분 : 2xcosx^2
    두번째미분 : (곱의미분적용)
    2cosx^2 - 4x^2sinx^2

  • @a-manthegeneral
    @a-manthegeneral ปีที่แล้ว +9

    13:05 I'm a CS major (junior btw) watching this lol
    These videos make me feel good lol

  • @Prism019
    @Prism019 ปีที่แล้ว +3

    31:38 You can hear the relief in that "Yes!" Congrats on getting a good take!

  • @stolenmonkey7477
    @stolenmonkey7477 8 หลายเดือนก่อน +2

    1:53 this was genuinely so funny I love that lol

  • @ivantolkachev4808
    @ivantolkachev4808 11 หลายเดือนก่อน +4

    For cos(2xh) - 1 you can also do the cos^2(xh) - sin^2(xh) -1 = -2sin^2(xh) to avoid the trig in the denominator

  • @velimir_ikalovic
    @velimir_ikalovic 11 หลายเดือนก่อน +1

    What amazes me is that I'm 47 yo, finished my highschool long time ago, dropped from university on second year, I don't use calculus in my life at all, and I still manage to understand most of this.

  • @mcalkis5771
    @mcalkis5771 ปีที่แล้ว +20

    Always a good day when you upload Steve. I always enjoy your videos where you do proofs like this.

  • @albertogarcia4177
    @albertogarcia4177 ปีที่แล้ว +2

    Seems all ok but this finding of the second derivative using the symmetric derivative as shown is good only for x≠0, see 26:57 where you multiply and divide for 2x. Would be need complete the proof for x=0, i guess is not hard, pluging x=0 in the initial steps, see the formula is also ok

  • @gallium-gonzollium
    @gallium-gonzollium ปีที่แล้ว +13

    Me when I try a calculation and I do it the more complicated way:

  • @nathanperkin1163
    @nathanperkin1163 ปีที่แล้ว +17

    I'm 15 and in year 11 (grade 10), and i haven't officially been taught calculus yet, but i find these kinds of videos super interesting!

    • @lirosphere956
      @lirosphere956 ปีที่แล้ว +5

      You're in for a treat if you go deep in this channel

    • @hattapalkan8395
      @hattapalkan8395 ปีที่แล้ว +1

      Proud of you brother. Keep it going

    • @idjles
      @idjles ปีที่แล้ว

      Calculus will keep you fascinated for the rest of your life - even when you are 80. Keep enjoying it.

    • @TheEGod.
      @TheEGod. ปีที่แล้ว

      im a little younger then you but i underatand very well. I remember I didnt understand them a year ago but I just couldnt stop watching these videos.

    • @diogr22
      @diogr22 ปีที่แล้ว

      same here bro

  • @argonwheatbelly637
    @argonwheatbelly637 11 หลายเดือนก่อน +14

    This is math candy. Awesome! ❤

    • @IloveUraniumSoMuch
      @IloveUraniumSoMuch 5 หลายเดือนก่อน

      Yo your comment is the thumbnail!

  • @gajamsai2957
    @gajamsai2957 8 หลายเดือนก่อน

    31:36 the excitement and happiness 👏😘 such happiness in his face very nice to see good answer sir 👍👏👏

  • @garyhuntress6871
    @garyhuntress6871 ปีที่แล้ว +3

    That was excellent!! I was on the edge of my seat!

  • @Veefencer
    @Veefencer 9 หลายเดือนก่อน

    The only thing I like more than trying to follow all of the steps you take during the video is your sheer joy when the thing is finally done! I can't help smiling as well. Thank you for reminding me that i really like math!)

  • @Dantido
    @Dantido ปีที่แล้ว +4

    Hey there.
    Before asking this, just wanted to say I love your videos. Thanks to you I've found out about my interest in math as a hobby, and I can't commend how interesting and satisfying stuff like calculus can be when you understand it enough.
    With that out of the way, I also wanted to ask you a question,
    which definition of derivative do you prefer?
    f(x-h) - f(x)
    lim ----------------
    x->0 h
    or
    f(x) - f(h)
    lim -------------
    x->h x - h

  • @potato4521
    @potato4521 10 หลายเดือนก่อน

    I was able to do it by applying the derivative twice so d/dx (sin(x^2)) = cos(x^2)*2x.
    Then d/dx (cos(x^2)*2x). using the product rule you get -sin(x^2)*2x*2x + cos(x^2)*2.
    This simplifies to -4x^2*sin(x^2) + 2*cos(x^2)

    • @theweirdwolf1877
      @theweirdwolf1877 5 หลายเดือนก่อน

      No shit sherlock but that’s not the point of the video

  • @kooshkooshyunger1438
    @kooshkooshyunger1438 ปีที่แล้ว +5

    You forgot to close the parentheses of the first d/dx at 30:27

  • @spudhead169
    @spudhead169 ปีที่แล้ว +6

    You've explored tetration a few times, but can the concept tetration be extended to include non-integers? If we define Tn(x) to be a tetration function such that T3(x) = x^x^x and T5(x) = x^x^x^x^x, then what happens if we plug in say 2.5 for n or even i? Is that even possible?

    • @ciple8330
      @ciple8330 ปีที่แล้ว

      I researched this a bit and it seems very interesting. For anyone interested, go to Wikipedia>Tetration>Extensions (go to the 'heights' section for n) and search up "Tetration Forum" if you want to see more discussion on tetration.

  • @m1n3c4rt
    @m1n3c4rt ปีที่แล้ว +30

    wow, your videos are so consistent that i didn't even notice this was from an hour ago
    also 25:30 funny integral sign :)

    • @TheZerovirus1000
      @TheZerovirus1000 ปีที่แล้ว

      ikr! I didn't notice until you pointed it out. I love this format

  • @indescribablecardinal6571
    @indescribablecardinal6571 11 หลายเดือนก่อน +1

    26:55 When both limits needed that 2x•2x/(2x•2x), and that finally connected with the answer given by chen lu, then I smirked a little c:

  • @Owen_loves_Butters
    @Owen_loves_Butters ปีที่แล้ว +1

    18:53 Actually, you can use L'Hôpital's as long as you don't derive d/dx[sinx]=cosx from it, and there are other ways to prove the derivative of sine is cosine :)

  • @kevinstreeter6943
    @kevinstreeter6943 ปีที่แล้ว

    Reminds me of the time when I was asked if there is 2 ways of solving a problem, which one should be used? I replied to use the easier method.

  • @BradMurray
    @BradMurray 8 หลายเดือนก่อน +1

    This was really beautiful; thank you!

  • @lukaskamin755
    @lukaskamin755 8 หลายเดือนก่อน +1

    5:30 and what if the graph is not symmetrical relative to the "angle" point, say if there's a sum or difference of a linear function and an abs function with various angle coefficients. What meaning does symmetrical derivative has, will it be relevant? It doesn't seem obvious

  • @ThsHunt
    @ThsHunt 8 หลายเดือนก่อน +1

    When teacher asks a simple question for 10 points

  • @oryx3
    @oryx3 ปีที่แล้ว +2

    I think we're going to need a bigger board 🦈

  • @AshX_human
    @AshX_human 11 หลายเดือนก่อน

    14:56 (x+h)^2 = x^2 + y^2 + 2xh, (x-h)^2 = x^2 + y^2 - 2xh, select alpha=x^2+y^2 and beta=2xh and you didn't need the sum to product formula :)

  • @DarkTouch
    @DarkTouch 8 หลายเดือนก่อน

    im never going to use the product rule and chain rule again because i loved doing it the long way. !!!! Nice proof.

  • @darktrinity9125
    @darktrinity9125 ปีที่แล้ว +16

    Now prove that limit is true by the definition of a limit (epsilon delta)

  • @joshuahillerup4290
    @joshuahillerup4290 ปีที่แล้ว

    It's been a long time since I took calc 1, but I'm now getting flashbacks to a bunch of weird limit definitions

  • @tortillajoe
    @tortillajoe ปีที่แล้ว +6

    Now do ε-δ 👀

  • @xyz.ijk.
    @xyz.ijk. ปีที่แล้ว +1

    Chain rule? What is the Chain rule? What happened to Chen lu? Did you mean Chen lu? I miss the old days. You are still a great teacher and always had a wonderful sense of humor.

  • @chhsel
    @chhsel 8 หลายเดือนก่อน +2

    some people enjoy using 2 sticks to start a fire....

  • @felipesb2
    @felipesb2 ปีที่แล้ว +1

    31:22 Hahaha I thought that I was the only one who get that excited when I finish a "uncrackable problem" like this one
    good job

  • @anandmoodley3787
    @anandmoodley3787 11 หลายเดือนก่อน +1

    That was brilliant!. As a student I enjoyed every bit of that

  • @dudl2945
    @dudl2945 ปีที่แล้ว +1

    I'm always looking for entertaining videos to watch while eating my food. Although I love maths, I would have never guessed I'll end up with this kind of stuff as best eating videos

  • @trelosyiaellinika
    @trelosyiaellinika ปีที่แล้ว +1

    Absolutely beautiful!

  • @actualRocketScientist
    @actualRocketScientist 10 วันที่ผ่านมา

    Love the marker drop at the end! Boom! Marker drop!

  • @SakretteAmamiya
    @SakretteAmamiya 11 หลายเดือนก่อน

    for the cos(2xh)-1, you can make it into -2sin(xh)^2 😀

  • @johnchestnut5340
    @johnchestnut5340 7 หลายเดือนก่อน

    I do appreciate it! Thank you. I am surprised at how much I remember. I am also keenly aware that I have forgotten so very much. But thank you for the videos!

  • @lol1991
    @lol1991 ปีที่แล้ว +2

    I literally solved the first derivative of this last saturday (by the definition)

  • @aymathconcoursprep
    @aymathconcoursprep 9 หลายเดือนก่อน

    this derivitive AkA fs(x) used also in Mechanics Element finite or Numerical methode to describe descritisation of grid points in the plan

  • @punitpasricha3876
    @punitpasricha3876 11 หลายเดือนก่อน +1

    We extremely appreciate your effort

  • @Diro_Nikhil
    @Diro_Nikhil 11 หลายเดือนก่อน

    Man never used the 2nd derivative by 1st principal…. This is so good

  • @viktor-kolyadenko
    @viktor-kolyadenko ปีที่แล้ว

    It's simple. Already the first derivative of this function “jumps” very much. For x> 2pi for example.

  • @sphakamisozondi
    @sphakamisozondi ปีที่แล้ว

    Bro did derivative calculus using 1st principles. I'm beyond impressed

  • @rockapedra1130
    @rockapedra1130 8 หลายเดือนก่อน

    This was endless excruciating pain followed by revelation. Sometimes you have to have faith and keep slogging! Wow! You did this as an 8 year old? Proof of alien visitation! I'm calling the History Channel!!!!!!

  • @HasanZekiAlp
    @HasanZekiAlp ปีที่แล้ว +1

    Hey, Guy, you’re great! While I was following you, frankly, I lost a few times, but, you delivered the ship to the quay, in sane!…

  • @samdean1966
    @samdean1966 2 หลายเดือนก่อน

    Fine and interesting but a little circular. You need to know the 2nd derivative EXISTS in order to apply the symmetric 2nd derivative. The way of knowing that is... pretty much the chain rule.

  • @joshuahillerup4290
    @joshuahillerup4290 ปีที่แล้ว

    I can't wait for the follow-up video

  • @michaelroy1631
    @michaelroy1631 ปีที่แล้ว +1

    very satisfying!

  • @joseph_soseph9611
    @joseph_soseph9611 8 วันที่ผ่านมา

    Your class was catching strays at the end. I'm sure some of them would also appreciate this

  • @alexdefoc6919
    @alexdefoc6919 ปีที่แล้ว +1

    i can aprove this worked. 30:08 before this. In my head ❤. I am really happy i can do it.

  • @TheZerovirus1000
    @TheZerovirus1000 ปีที่แล้ว +1

    31:37 certified mic drop / pen drop moment

  • @Mandq.
    @Mandq. ปีที่แล้ว +1

    if anyone used chain rule and producst rule the answer would be: 2( -x^3 * sin(x^2) + cos(x^2) )

  • @Amoeby
    @Amoeby ปีที่แล้ว

    When I saw that we need to find (sin(x^2))'' I was thinking how is this video this long and then BRP pulled out the limit definition of the derivative.

  • @charlievane
    @charlievane ปีที่แล้ว +1

    31:36 happiest man alive 😂

  • @Annnabannanna
    @Annnabannanna ปีที่แล้ว

    Omg, grea ttiming, I have an exam in 2 days on calculus.

  • @eggthepro2472
    @eggthepro2472 ปีที่แล้ว +1

    17:07 After the 4th line, I know why he factored out the 2sin(x^2) but i don’t necessarily get how. The 2nd term in that question doesn’t have the factor sin(x^2)?? am i just reading it wrong
    Also, I don’t understand why he uses the symmetric derivative for this function to begin with. I don’t quite know if it’s easier, but the function itself has a derivative with the domain of all real numbers, so wouldn’t it be easier to just take the derivative of that again?? i think he said that the symmetric derivative doesn’t imply the existence of the first derivative but if the first derivative does exist then what’s the point?

    • @jb31842
      @jb31842 ปีที่แล้ว

      It's not the 2nd term in the numerator, but the 3rd term (which is where the "-1" then came from).
      As he explained at the beginning, if you know the normal derivative exists, then the symmetric derivative also exists, and the two are equal. Since you can choose either representation, evidently the symmetric derivative was easier to work with.

    • @eggthepro2472
      @eggthepro2472 ปีที่แล้ว

      @@jb31842haha i totally just didn’t see that at all, thanks for pointing that out

  • @WingedShell82
    @WingedShell82 11 หลายเดือนก่อน +1

    That was a very fun 32 minutes of my life. Honestly, this makes me want to practice doing this kind of stuff on my own just because it looks fun lol.

    • @EmpyreanLightASMR
      @EmpyreanLightASMR 11 หลายเดือนก่อน

      It's fun when you are seeing the steps you have to take. Not so much when you're standing at a crossroads. Though I learned earlier today that sometimes you just have to make a decision (a big decision, as Bob Ross would say) and stick with it until you hit a dead end.

  • @scottleung9587
    @scottleung9587 ปีที่แล้ว +2

    Very cool!

  • @alexprospal8548
    @alexprospal8548 11 หลายเดือนก่อน

    I apreciate it. Thank you. We all apreciate you.

  • @earl8295
    @earl8295 ปีที่แล้ว +7

    Blackpenredpen be like: Im going to prove this using Epsilon Delta defintion

  • @jacquesduranceau8762
    @jacquesduranceau8762 11 หลายเดือนก่อน

    My students used to keep a catalog of the outrageous things I said over the year. I think "just give up" and "don't be happy to quickly" satisfy the criteria!. I have one question: Doesn't the "'symmetric derivative" (which looks akin to a central finite difference) lead to f' = 0 for absolute value at the critical point?

  • @probablemente_sebas6612
    @probablemente_sebas6612 11 หลายเดือนก่อน

    OH MY GOD we had the exact same expression of the symmetric derivative as homework and the teacher never told us it was an actual definition 😭.

  • @lore7_03
    @lore7_03 11 หลายเดือนก่อน

    "i'm not gonna prove that limit, I'm just gonna use it"
    proceeds to prove that limit

  • @black-shadow5715
    @black-shadow5715 ปีที่แล้ว +8

    Can we use the following definition to calculus the second derivative of sin(x²) ?
    f''(x) =
    lim h->0 [ (f(x+2h)-2f(x+h)+f(x)) / (h²) ]
    I don't know if using this formula would be more easier than the other one in the video, I don't have the energy to try it 😂
    And also, I don't know if the formula is right or not but I do a little demonstration to proove it and I think the formula is right...
    Please tell me 👀
    PS : sorry for my english, I'm actually french

    • @thenew3dworldfan
      @thenew3dworldfan ปีที่แล้ว

      Definitely, in fact this might be there regular second derivative not the symmetric one, which might exist even if the function isn’t twice differentiable?

  • @tobybartels8426
    @tobybartels8426 ปีที่แล้ว

    Proving that the second symmetric derivative equals the second derivative, on the assumption that the second derivative exists, is not too bad; the function must be differentiable on a neighbourhood of the point, so you can use L'Hôpital's Rule. That reduces it to the symmetric derivative of the derivative.
    However, proving that the symmetric derivative equals the derivative, on the assumption that the derivative exists, is harder, because it might not be differentiable anywhere else, so you can't use L'Hôpital's Rule. Instead, you can use the same kind of tricks that you'd use to prove the Chain Rule.

  • @EliteCubingAlliance
    @EliteCubingAlliance ปีที่แล้ว

    13:11 "If you're also a calculus teacher, you know what to put on the final exam"
    Don't give my professor any ideas 😂

  • @shubhamsaini4871
    @shubhamsaini4871 ปีที่แล้ว

    How to check if we can integrate a function or not?.
    Like e^cosx is non integratable so how do we know that it is non integratable

  • @sumedh-girish
    @sumedh-girish 6 หลายเดือนก่อน

    Buy this man a bigger board

  • @General12th
    @General12th ปีที่แล้ว +1

    So good!

  • @toopytoopy8547
    @toopytoopy8547 11 หลายเดือนก่อน

    maybe using the fact that sin(x²)=Im(exp(ix²)) helps as d²sin(x²)/dx² = Im(d²(exp(ix²))/dx²)
    And with your method, exp(i(x+h)²)+exp(i(x-h)²)-2exp(ix²) = exp(i(x²+h²))[exp(2ixh)+exp(-2ixh)]-2exp(ix²)= 2exp(ix²)[cos(2xh) exp(ih²)-1]. Then, we have [cos(2xh) exp(ih²)-1]/h² = exp(ih²)[cos(2xh)-1]/h²+[exp(ih²)-1]/h². For the first part, we know that lim [cos(2xh)-1]/h² = -4x²*lim [1-cos(2xh)]/(2xh)² = -2x² and lim exp(ih²) = 1. For the second part, lim [exp(ih²)-1]/h² = i (as h²->0 and lim [exp(ih)-1]/h=iexp'(0)=i). Thus, lim [cos(2xh) exp(ih²)-1]/h² = -2x²+i . Therefore, d²(exp(ix²))/dx²=(-4x²+2i)exp(ix²) and finally, by taking the imagenary part, d²sin(x²)/dx² = -4x²sin(x²)+2cos(x²)

  • @EmpyreanLightASMR
    @EmpyreanLightASMR 11 หลายเดือนก่อน +1

    Everyone is inspired by this great video, but apparently I'm the only one who heard two cries for help. Twice! you asked for a bigger house to fit a bigger board. Someone get this man a larger house!

    • @blackpenredpen
      @blackpenredpen  11 หลายเดือนก่อน +1

      Thanks! I appreciate that. 😃

  • @huethehue
    @huethehue 5 หลายเดือนก่อน

    17:00
    why can you factor out sin(x^2) if it isn’t in the second term

  • @s1mppeli
    @s1mppeli 11 หลายเดือนก่อน

    This is the sort of exam question where you save time by just including a proof of the chain rule at the start. Just so you can use it to solve the problem ;Dd

  • @omintionpg3d652
    @omintionpg3d652 4 หลายเดือนก่อน

    yeah yeah im going to know the symmetric differentiation before the chain and product rule

  • @lornacy
    @lornacy 9 หลายเดือนก่อน

    I will have to watch this more than one time!

  • @PapaCarlo87
    @PapaCarlo87 11 หลายเดือนก่อน +1

    Actually it took me only 16 minutes, while watching your video at x2 speed.

  • @genocider5868
    @genocider5868 ปีที่แล้ว +1

    wait, does it mean you can just use symmetric derivative as a normal derivative?
    Or is it only second symmetric derivative = second derivative?
    Or is it just coincidence?

    • @blackpenredpen
      @blackpenredpen  ปีที่แล้ว +2

      If the function is differentiable in the usual sense, then yes. Symmetric derivative = regular derivative

    • @Memzys
      @Memzys ปีที่แล้ว +1

      you can use symmetric derivative definition in place of the regular derivative, they should be the same, except sometimes the symmetric derivative will you give you an answer while the regular derivative does not (when there is a sharp turn in the graph, like f(2) = |x - 2|)
      quoting wikipedia, "If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true"
      this applies for any nth derivative, not just the 2nd

  • @EC4U2C_Studioz
    @EC4U2C_Studioz ปีที่แล้ว +1

    Looks like a complete nightmare using the definition of the derivative to solve that instead of using the derivative rules.

  • @askandpushpaltiwary8537
    @askandpushpaltiwary8537 ปีที่แล้ว +1

    appreciate the hard work!

  • @phat5340
    @phat5340 ปีที่แล้ว

    Why not use mvt

  • @Deniz-zd2nx
    @Deniz-zd2nx ปีที่แล้ว

    Can you find the complex solution of 3^x+3=30 of course if there is

    • @xxxgetr3ktxxx740
      @xxxgetr3ktxxx740 ปีที่แล้ว

      3^x + 3 = 30
      3^x = 27
      x = log_3(27)
      x = ln(27)/ln(3)
      x = 3ln(3)/ln(3)
      x = 3
      Solving the normal way, we'd get only 3 but we can get the complex solutions when using Euler's formula to get 27 = 27e^i2nπ
      Plugging back in:
      x = ln(27e^i2nπ)/ln(3)
      x = (ln(27) + ln(e^i2nπ))/ln(3)
      x = (3ln(3) + i2nπ)/ln(3)
      x = 3 + (2nπ/ln3) i, n∈Z

  • @gietie1694
    @gietie1694 ปีที่แล้ว

    its beautiful

  • @mohamed.chakib_
    @mohamed.chakib_ ปีที่แล้ว

    100 double intégral and 100 triple intégral ❤